## Σάββατο, 3 Μαΐου 2014

### ORTHOLOGIC - PERSPECTIVE TRIANGLES

Let ABC be a triangle and A'B'C' the cevian triangle of P.

Denote:

Ab, Ac = The circumcenters of APB', APC', resp.

Bc, Ba = The circumcenters of BPC', BPA', resp.

Ca, Cb = The circumcenters of CPA', CPB', resp.

1. M1a,M1b,M1c = The midpoints of AbAc,BcBa,CaCb, resp.

Which is the locus of P such that:

1.1. ABC, M1aM1bM1c are perspective?

1.2. ABC, M1aM1bM1c are orthologic?

1.3. The perpendicular bisectors of AbAc,BcBa,CaCb are concurrent?

For P = G:

1.2. ABC, M1aM1bM1c are orthologic.

Orthologic center (M1aM1bM1c, ABC) = N

Orthologic center (ABC, M1aM1bM1c) : Anopolis #1284, #1295

1.3. The perpendicular bisectors concur at van Lamoen Circle Center X(1153)

2. M2a,M2b,M2c = The midpoints of BcCb, CaAc, AbBa, resp.

Which is the locus of P such that:

2.1. ABC, M2aM2bM2c are perspective?

2.2. ABC, M2aM2bM2c are orthologic?

2.3. The perpendicular bisectors of BcCb, CaAc, AbBa are concurrent?

For P = G:

2.2. ABC, M2aM2bM2c are orthologic.

Orthologic center (M2aM2bM2c, ABC) = ?

Orthologic center (ABC, M2aM2bM2c) = G

2.3. The perpendicular bisectors concur at van Lamoen Circle Center X(1153)

3. M3a,M3b,M3c = The midpoints of BaCa, CbAb, AcBc, resp.

Which is the locus of P such that:

3.1. ABC, M3aM3bM3c are perspective?

3.2. ABC, M3aM3bM3c are orthologic?

3.3. The perpendicular bisectors of BaCa, CbAb, AcBc are concurrent?

For P = G

3.2. ABC, M3aM3bM3c are orthologic.

Orthologic center (M3aM3bM3c, ABC) = O

Orthologic center (ABC, M3aM3bM3c) = ?

3.3. The perpendicular bisectors concur at van Lamoen Circle Center X(1153)

4. Which is the locus of P such that:

4.1. M1aM1bM1c, M2aM2bM2c

4.2. M1aM1bM1c, M3aM3bM3c

4.3. M2aM2bM2c, M3aM3bM3c

are perspective/orthologic ?

4.4. The Euler lines of M1aM1bM1c, M2aM2bM2c, M3aM3bM3c are concurrent?

For P = G ??

5. Which is the locus of P such that:

4.1. M1aM2aM3a, M1bM2bM3b

4.2. M1aM2aM3a, M1cM2cM3c

4.3. M1bM2bM3b, M1cM2cM3c

are perspective/orthologic ?

4.4. The Euler lines of M1aM2aM3a, M1bM2bM3b, M1cM2cM3c are concurrent?

For P = G ??

Antreas P. Hatzipolakis, 4 May 2014